Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (x² + x) / (x⁴ - 3x² - 4) dx

Accepted Answer

First, factor the denominator $$x^{4}$$ - $$3x^{2} - 4$. Recognize it as a quadratic in terms of x$$^{2}$$: let u$$ = $$x^{2}$$, so the expression becomes $$u^{2} - 3u - 4$. Factor this quadratic to get (u - 4)(u + 1$$)$$, which translates back to (x$$^{2}$$ - 4)(x$$^{2}$$ + 1). $$Next, factor $$x^{2} - 4$ further as a difference of squares: (x - 2)(x + 2$$)$$. So the full factorization of the denominator is (x - 2)(x$$ + $$2)(x^{2} + 1$$)$$. Set up the partial fraction decomposition for the integrand $$\frac{$$x^{2} + x$$}{(x - 2)(x + $$2)(x^{2} + 1$$)}$$ as follows:

$$\frac{$$x^{2} + x$$}{(x - 2)(x + $$2)(x^{2} + 1$$)} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{Cx + D}{$$x^{2} + 1$$}$$,

where A$, B$, C$, and D$ are constants to be determined. Multiply both sides of the equation by the denominator (x - 2)(x$$ + $$2)(x^{2} + 1$$)$$ to clear the fractions, resulting in:

x$$^{2}$$ + x = A(x + 2)(x$$^{2}$$ + 1) + B(x - 2)(x$$^{2}$$ + 1) + (Cx + D)(x - 2)(x + 2). $$Expand the right-hand side and collect like terms in powers of $$x. $$Then, equate the coefficients of corresponding powers of $$x on $$both sides to form a system of equations. Solve this system to find the values of $$A$$, $$B$$, $$C$$, and $$D. $$Once found, rewrite the integral as a sum of simpler integrals corresponding to each partial fraction term, which can then be integrated using standard techniques.

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx

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Key Concepts

Partial Fraction Decomposition

Factoring Polynomials

Integration of Rational Functions